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# Enjoyment of Mathematics: Selections from Mathematics for the Amateur

OTTO TOEPLITZ
TRANSLATED BY HERBERT ZUCKERMAN
Pages: 214
Stable URL: http://www.jstor.org/stable/j.ctt183pvvh

1. Front Matter (pp. i-iv)
2. Preface (pp. v-vi)
4. Introduction (pp. 5-8)

Mathematics, because of its language and notation and its odd-looking special symbols, is closed off from the surrounding world as by a high wall. What goes on behind that wall is, for the most part, a secret to the layman. He thinks of dull uninspiring numbers, of a lifeless mechanism which functions according to laws of inescapable necessity. On the other hand, that wall very often limits the view of him who stays within. He is prone to measure all mathematical things with a special yardstick and he prides himself that nothing profane shall enter his realm.

Is it possible...

5. 1. The Sequence of Prime Numbers (pp. 9-13)

6 is equal to 2 times 3, but 7 cannot be similarly written as a product of factors. Therefore 7 is called aprime or prime number. A prime is a positive whole number which cannot be written as the product of two smaller factors. 5 and 3 are primes but 4 and 12 are not since we have 4 = 2·2 and 12 = 3·4. Numbers which can be factored like 4 and 12 are calledcomposite. The number 1 is not composite but, because it behaves so differently from other numbers, it is not usually considered a prime...

6. 2. Traversing Nets of Curves (pp. 13-17)

A streetcar company decides to reorganize its system of routes without changing the existing tracks. It wishes to do this in such a way that each section of track will be used by just one route. Passengers will be allowed to transfer from route to route until they finally reach their destinations. The problem is:how many routes must the company operate in order to serve all sections without having more than one route on any section?

For a very small city with car lines as in Fig. 1, the problem is quite simple. One route could go fromA...

7. 3. Some Maximum Problems (pp. 17-22)

1. Let us compare the areas of various rectangles of two-inch perimeter. Some are shown in Fig. 5. The width of each rectangle must be less than 1 inch, and the closer it is to 1 inch the smaller is the height and also the area. If the height is close to 1 inch, then the width and again the area are very small. The intermediate rectangles have larger area, and one might ask which of the rectangles has the largest area. This is a maximum problem. It is probably the simplest and oldest of all such problems, and so perhaps...

8. 4. Incommensurable Segments and Irrational Numbers (pp. 22-26)

The measurement of length, area, and volume is at the root of all geometry. To measure one line segment by another, we see how many times one goes into the other. This is simple enough if it goes in exactly without leaving a remainder. If the smaller segment does not go into the larger one exactly, then we look at the remainder. It may happen that the remainder is one-half, one-third, two-thirds, or some other similar fractional part of the segment we are using as a measure. If so, we have a sort of substitute measure, a fractional part of...

9. 5. A Minimum Property of the Pedal Triangle (pp. 27-30)

We shall again consider a problem of the kind discussed in Chapter 3, but this time it might more properly be called a minimum problem. It will serve to introduce mathematical methods that are highly refined yet clear and simple. The theorem and the proof we shall give here are the work of H. A. Schwarz. Although the theorem is only a relatively minor mathematical problem, it shows how this great mathematician’s genius manifests itself, equally in relatively trivial and extremely significant work.

1. Before considering our main theorem, let us look at a very simple problem concerned with the law...

10. 6. A Second Proof of the Same Minimum Property (pp. 30-34)

1. In the last chapter we proved that of all the triangles inscribed in a given acute-angle triangle, the pedal triangle has the smallest perimeter. It is worthwhile to consider another proof of the same theorem, because this second proof will illustrate some new ideas and, for our purposes, the methods used are of more importance and interest than the mere mathematical content of new theorems. The previous proof, originally given by H. A. Schwarz, depended essentially on the fact that a straight line is the shortest distance between two points, and it made use of the idea of reflection of...

11. 7. The Theory of Sets (pp. 34-42)

The subject of this chapter lies at the very foundations of mathematics. However, our interest in it will depend more on the beauty and simplicity of the manner in which it is built up than on its significance for mathematics in general. The theory of sets, originated by Georg Cantor, is a truly mathematical theory which starts with only the very simplest concepts and builds up to a ramified and important subject through the use of pure reasoning.

Are there more whole numbers than even numbers? Which are more numerous, the points of a line segment or the points of...

12. 8. Some Combinatorial Problems (pp. 43-51)

1. A simple example will serve to show what type of problem we shall discuss. Suppose we have 4 red balls (R), 1 yellow ball (Y), and 2 white balls (W). These balls are supposed to be of the same size and weight and completely indistinguishable except by their colors. We also suppose that we have two urns,AandB. UrnAwill hold exactly 3 balls,Bwill hold 4.In how many different ways can the7colored balls be distributed between the urns A and B?

Since we have a very simple case with only two urns,...

13. 9. On Waring’s Problem (pp. 52-61)

The sequence of squares$1,4,9,16,25,\cdots$becomes less and less dense as we go further out. The gaps between the consecutive squares become longer and longer. Although many numbers are not squares, some of them can at least be considered as sums oftwosquares, for example, 13 = 9 + 4, 41 = 25 + 16, etc. But not every number can be written as a sum of two squares. If we try to express the number 6 as a sum of two squares, the available squares are 1 and 4, the only squares that are less than 6. Neither...

14. 10. On Closed Self-Intersecting Curves (pp. 61-66)

1. The curves that we shall discuss in this chapter are of a special kind. Although they may be quite complicated, they must satisfy certain conditions. First, they must be traversible in a single passage. That is, one should be able to draw the whole curve with a single stroke, starting at a given point and never taking the pencil from the paper until the curve is completely drawn. Second, they must be closed. That is, when one draws the curve he should be able to start at a given point, trace out the whole curve, and return to the original...

15. 11. Is the Factorization of a Number into Prime Factors Unique? (pp. 66-73)

Starting with any given number, one can keep splitting it into factors until one finally has only prime factors. For example, 60 may be factored as 6 · 10, 6 as 2 · 3, and 10 as 2 · 5, so that we finally have

60 = 2 · 3 · 2 · 5

and these four factors are all prime.

Still using the example 60, we could first have factored it as 60 = 4 · 15, 4 = 2 · 2, 15 = 3 · 5, from which we have

60 = 2· 2 · 3 · 5....

16. 12. The Four-Color Problem (pp. 73-82)

1. In 1879 Cayley discussed the following problem. A map is usually printed in several colors in order to distinguish between the different countries. It would be best if each country were printed in a different color, but this is too costly. Instead it is customary to use as few colors as possible, being careful that countries are always differently colored when they are next to each other. Fig. 35a represents the map of an island that requires three colors, blue for the sea and two colors for the two countries. Fig. 35b requires four colors. The three countries all touch...

17. 13. The Regular Polyhedrons (pp. 82-88)

1. We are going to make use of Euler’s theorem to obtain an entirely different sort of result. We shall investigate the question: Do regular polyhedrons exist, and how many are there? A polyhedron is a solid figure bounded by portions of planes called faces. According to Euclid, a polyhedron is “regular” if all its faces are congruent polygons having equal sides and angles (regular polygons).

Our problem will be more extensive and the answer more satisfactory if we use a more general definition. We shall say that a polyhedron is “regular” if all its faces have the same number of...

18. 14. Pythagorean Numbers and Fermat’s Theorem (pp. 88-95)

1. According to the Pythagorean theorem, the square on the hypotenuse of a right triangle has the same area as the sum of the squares on the two legs. Conversely, if three line segments are such that the square on one is equal to the sum of the squares on the other two, then the three segments will form a right triangle. The equationa2+b2=c2represents the fact that the segments of lengtha,b,care the sides of a right triangle.

We have already seen in Chapter 4 that the hypotenuse and legs of an...

19. 15. The Theorem of the Arithmetic and Geometric Means (pp. 95-103)

A careful experimenter measures a certain object and finds its length to be 2.172 feet. On repeating his measurements twice more he obtains the lengths 2.176 ft. and 2.171 ft. What should he accept as the true length of the object? In such a case it is customary to use the average of the measurements, to add them and divide by their number. The experimenter would find the total to be 6.519 and, dividing by 3, would accept 2.173 ft. as the length of the object. This average that we have described is called the arithmetic mean. The arithmetic mean...

20. 16. The Spanning Circle of a Finite Set of Points (pp. 103-110)

1. We consider a finite set consisting ofnpoints${{P}_{1}},{{P}_{2}},\cdots ,{{P}_{n}}$, in a plane. The distances between each pair of pointsPi, andPj, can be measured. There must be a largest distance among this finite¹ set of distances. This largest distance is called the “span” of the set of points.

If a set ofnpoints has spand, then we can draw a circle of radiusdthat completely surrounds thenpoints (Fig. 50). All we need do is draw a circle of radiusdwith any one of thenpoints, sayP1, as center. Since...

21. 17. Approximating Irrational Numbers by Means of Rational Numbers (pp. 111-119)

The value$\frac{22}{7}$is an old and very familiar approximate value forπ, the area of a circle of radius 1. Also √2 is nearly$\frac{7}{5}$. Precisely what do we mean by such statements as these? The words “approximate” and “nearly” do not have a real place in mathematical speech, yet these statements must have some significance. Why is$\frac{22}{7}$invariably used to approximateπ, in preference, say, to a fraction with denominator 8?

1. If any numberwis given, then fractions or (as a mathematician would say) rational numbers that are arbitrarily close to it can be found....

22. 18. Producing Rectilinear Motion by Means of Linkages (pp. 119-129)

James Watt’s original steam engine was equipped with a remarkable mechanism called Watt’s parallelogram because of its shape. The apparatus, which is shown schematically in Fig. 58, consists of five rods hinged together atC,D,E,F. AtAandBthe rods are held in place by means of pivots that allow the rods to rotate.¹ All the hinges are made in such a way that the rods can never move out of the plane. The piston rod is attached atF. The purpose of this apparatus was to force the end of the piston rod to move...

23. 19. Perfect Numbers (pp. 129-135)

Book IX of Euclid’sElementsis the third and last book that is occupied with arithmetic. This book includes the proof of the infinitude of prime numbers, which we have reproduced in Chapter 1, and it concludes with a discussion of so-called perfect numbers. Perfect numbers are also mentioned by Plato, especially in an enigmatic passage in hisRepublicwhere, in an obscure discussion of eugenics, he introduces the “nuptial number”.

The subject of perfect numbers and the theorems that were later proved concerning them are now no more than an interesting curiosity in the body of modern mathematics. But...

24. 20. Euler’s Proof of the Infinitude of the Prime Numbers (pp. 135-139)

Euclid’s proof of the infinitude of the primes, which we discussed in Chapter 1, immediately precedes his consideration of perfect numbers. Euler, who took up and extended the study of perfect numbers, produced another proof of the infinitude of primes. This proof uses the same ideas that are basic in the theory of perfect numbers.

We must make two simple observations before proceeding with the proof.

1. LetABbe a line segment 2 feet long.

If we traverse it fromAto its midpointM, from there to the midpointM1of the remainderMB, from there to the midpoint...

25. 21. Fundamental Principles of Maximum Problems (pp. 139-142)

We have repeatedly discussed maximum problems. In Chapters 3, 5, and 6 we exhibited some mathematical miniatures which mathematicians of great ability have found time to produce along with their more important and lengthier work. In this chapter we shall discuss some principles that are basic to all these problems.

These principles can be developed by considering an extremely simple maximum problem.Let a triangle be given(it is best to think of it as cut out of paper). The problem is tofind the two points P and Q that are as far apart as possible on the surface...

26. 22. The Figure of Greatest Area with a Given Perimeter (pp. 142-146)

Why do soap bubbles have the shape of a sphere? It is because the walls are made of a substance that is subject to cohesive forces tending to increase the thickness and decrease the area of the walls. The pressure of the air does not come into play, but the enclosed air maintains a fixed volume while the area becomes as small as possible. The soap bubble solves the problem of finding the solid figure with given volume that has the least area.

The problem that we shall solve is more modest than the one which is solved by every...

27. 23. Periodic Decimal Fractions (pp. 147-160)

1. The expansion of a common fraction into a decimal is a familiar process. When it is carried out, decimals of quite different sorts may arise, as is shown by the following examples:$\begin{array}{*{35}{rll}} \text{I}. & \frac{1}{5}=0.2\quad \quad \quad \, , & \frac{3}{40}=0.075\quad \quad \quad \quad \quad \ \ \, , \\ \text{II}. & \frac{4}{9}=0.4444\cdots , & \frac{1}{7}=0.142857142857\cdots , \\ \text{III}. & \frac{1}{6}=0.1666\cdots , & \frac{7}{30}=0.2333\cdots \quad \quad \quad \ \ \, . \\ \end{array}$

The simplest are those of type I. Remembering the meaning of a decimal, we can write them as common fractions with their denominator powers of 10:$\frac{1}{5}=\frac{2}{10},\frac{3}{40}=\frac{75}{1000}.$

These equations do not imply anything unusual. They merely assert that the given fractions can be “extended” so that their denominators become powers of 10. That can be done with any fraction whose denominator divides some power of 10. The...

28. 24. A Characteristic Property of the Circle (pp. 160-163)

When it rains, the ground is wet; when the ground is wet, it is not necessarily raining. This example is frequently used to elucidate the difference between a theorem and its converse. Clear as it may be in this formulation, it is very badly confused in ordinary life. Intelligent persons to whom the difference is crystal clear when it is brought to their attention are prone to mix it up unconsciously in ordinary intercourse. A political orator can often take a statement of his opponent and make it sound ridiculous by stating it in its converse form, without the trick’s...

29. 25. Curves of Constant Breadth (pp. 163-177)

1. A circle is defined as the curve all of whose points lie at a given distance from a fixed point, the center. The wheel is a direct practical application of this property of the circle. The hub of the wheel is held at a fixed height above the ground by the spokes of equal length, thus maintaining a smooth horizontal motion. In moving very heavy loads, the wheel and axle is sometimes not sufficiently strong. In this case one often resorts to the more primitive use of rollers. The load is merely rolled along over cylindrical rollers (Fig. 88) which...

30. 26. The Indispensability of the Compass for the Constructions of Elementary Geometry (pp. 177-187)

1. The constructions of elementary geometry are all carried out with the aid of a straightedge and compass. In fact, a distinguishing property of elementary geometry is the fact that the only implements allowed are the compass and straightedge. But these two instruments are not entirely necessary. There are many constructions in which one or the other can be dispensed with. More than this, according to the investigations of Mascheroni and the recently found earlier work of Mohr, the straightedge can be dispensed with entirely. All constructions that are possible with a straightedge and compass can be made with a compass...

31. 27. A Property of the Number 30 (pp. 187-192)

Neither 10 nor 21 is a prime number. But 10 = 2·5 and 21=3 · 7 have no divisor that is common to them both. For this reason they are called “relatively prime” numbers. The numbers 6 and 10 are not relatively prime; they have the common divisor 2.

Of all the numbers from 1 to 9, the numbers 3, 7, 9 are relatively prime to 10. Although 9 is relatively prime to 10, it is not a prime number. In the case of 12 the situation is different. Of the numbers from 1 to 11, only 5, 7, 11...

32. 28. An Improved Inequality (pp. 192-196)

In Chapter 27 we mentioned that Bonse’s proof of (8) actually gives the better inequality (9). As a matter of fact, the addition of one simple idea will allow us to prove even a little more in one direction, although, as we shall see, we will lose something in another direction.

The new idea is this: IfMis a number of the form 6m- 1 (a multiple of 6 decreased by 1), then in the decomposition ofMinto prime factors there must appear at least one prime which is also of the same form, 6x- 1....

33. Notes and Remarks (pp. 197-205)